Accuracy improvement in CORDIC through precomputation of the error bias

ABSTRACT

Performing a calculation using a coordinate rotation digital computer (CORDIC) algorithm. Execution of the CORDIC algorithm is begun. An error introduced by a truncated vector as a result of executing the CORDIC algorithm is pre-computed. The error is incorporated into a subsequent iteration of the CORDIC algorithm. Execution of the CORDIC algorithm is completed. The result of the CORDIC algorithm is stored.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates generally to an improved data processingsystem and in particular to devices and methods for performingcalculations more efficiently. Still more particularly, the presentinvention relates to improved techniques for more efficient calculationof coordinate rotation digital computer (CORDIC) algorithm rotationangles.

2. Description of the Related Art

A problem often confronted in computing is how to implement calculationof mathematical functions. Many mathematical functions cannot becalculated directly, but rather numerical techniques are employed toapproximate answers to the mathematical function.

One technique for solving mathematical functions on a computer is knownas the coordinate rotation digital computer (CORDIC) algorithm. TheCORDIC algorithm is an iterative arithmetic algorithm. The CORDICalgorithm is an efficient method for computing many complex mathematicalfunctions, such as, but not limited to, lattice filters, QRfactorizations, sine functions, cosine functions, fast Fouriertransformations, and generally transcendental functions that cannot beconstructed from a finite combination of constant algebraic,exponential, logarithmic functions and field operations. The CORDICalgorithm is commonly used in digital signal processing and in graphicalprocessing.

The basic task performed by the CORDIC algorithm is to rotate a 2 by 1vector through an angle using a linear, circular, or hyperboliccoordinate system. The CORDIC algorithm rotates the vector through asequence of elementary angles whose algebraic sum approximates thedesired rotation angle. These elementary angles have the property thatthe vector rotation through each of them may be computed easily with asingle shift and add operation. This formulation of CORDIC algorithmleads to a unified procedure to compute a wide range of complexmathematical functions using a fixed number of shifts and add computingsteps.

However, the CORDIC algorithm is an inherently inaccurate calculationhaving three main error types introduced from the quantization effectsof implementing the CORDIC algorithm. The first type of error occursfrom the quantization of the rotation angle. The second type of errorarises from rounding numbers of the finite precision arithmetic usedwhen executing the CORDIC algorithm. The third type of error arises fromthe rotation angle quantization, which arises from the rotation anglebeing incorrectly represented by the wrong sign.

SUMMARY OF THE INVENTION

The illustrative embodiments described herein provide for a method,computer usable program code and data processing system for performing acalculation using a coordinate rotation digital computer (CORDIC)algorithm. Execution of the CORDIC algorithm is begun. An errorintroduced by a truncated vector as a result of executing the CORDICalgorithm is pre-computed. The error is incorporated into a subsequentiteration of the CORDIC algorithm. Execution of the CORDIC algorithm iscompleted. The result of the CORDIC algorithm is stored.

BRIEF DESCRIPTION OF THE DRAWINGS

The novel features believed characteristic of the invention are setforth in the appended claims. The invention itself, however, as well asa preferred mode of use, further objectives and advantages thereof, willbest be understood by reference to the following detailed description ofan illustrative embodiment when read in conjunction with theaccompanying drawings, wherein:

FIG. 1 is a diagram of a data processing system, in accordance with anillustrative embodiment;

FIG. 2 is a flowchart of a prior art CORDIC algorithm in which aspectsof the present invention may be implemented;

FIG. 3 is a flowchart of an improved CORDIC algorithm, in accordancewith an illustrative embodiment;

FIG. 4 is a table comparing errors generated using a prior art CORDICalgorithm versus using the improved CORDIC algorithm shown in FIG. 3, inaccordance with an illustrative embodiment; and

FIG. 5 is a table comparing errors generated using a prior art CORDICalgorithm versus using the improved CORDIC algorithm shown in FIG. 4, inaccordance with an illustrative embodiment.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT

FIG. 1 is a diagram of a data processing system, in accordance with anillustrative embodiment. In an illustrative example, data processingsystem 100 includes communications fabric 102, which providescommunications between processor unit 104, memory 106, persistentstorage 108, communications unit 110, input/output (I/O) unit 112, anddisplay 114.

Processor unit 104 serves to execute instructions for software that maybe loaded into memory 106. Processor unit 104 may be a set of one ormore processors or may be a multi-processor core, depending on theparticular implementation. Further, processor unit 104 may beimplemented using one or more heterogeneous processor systems in which amain processor is present with secondary processors on a single chip. Asanother illustrative example, processor unit 104 may be a symmetricmulti-processor system containing multiple processors of the same type.

Memory 106 may be, for example, a random access memory or other form ofvolatile memory. Persistent storage 108 may take various forms dependingon the particular implementation. For example, persistent storage 108may contain one or more components or devices. For example, persistentstorage 108 may be a hard drive, a flash memory, a rewritable opticaldisk, a rewritable magnetic tape, or some combination of the above. Themedia used by persistent storage 108 also may be removable. For example,a removable hard drive may be used for persistent storage 108.

Communications unit 110, in these examples, provides for communicationswith other data processing systems or devices. In these examples,communications unit 110 is a network interface card. Communications unit110 may provide communications through the use of either or bothphysical and wireless communications links.

Input/output unit 112 allows for input and output of data with otherdevices that may be connected to data processing system 100. Forexample, input/output unit 112 may provide a connection for user inputthrough a keyboard and mouse. Further, input/output unit 112 may sendoutput to a printer. Display 114 provides a mechanism to displayinformation to a user.

Instructions for the operating system and applications or programs arelocated on persistent storage 108. These instructions may be loaded intomemory 106 for execution by processor unit 104. The processes of thedifferent embodiments may be performed by processor unit 104 usingcomputer implemented instructions, which may be located in a memory,such as memory 106. These instructions are referred to as program code,computer usable program code, or computer readable program code that maybe read and executed by a processor in processor unit 104. The programcode in the different embodiments may be embodied on different physicalor tangible computer readable media, such as memory 106 or persistentstorage 108.

Program code 116 is located in a functional form on computer readablemedia 118 and may be loaded onto or transferred to data processingsystem 100 for execution by processor unit 104. Program code 116 andcomputer readable media 118 form computer program product 120 in theseexamples. In one example, computer readable media 118 may be in atangible form, such as, for example, an optical or magnetic disc that isinserted or placed into a drive or other device that is part ofpersistent storage 108 for transfer onto a storage device, such as ahard drive that is part of persistent storage 108. In a tangible form,computer readable media 118 also may take the form of a persistentstorage, such as a hard drive or a flash memory that is connected todata processing system 100. The tangible form of computer readable media118 is also referred to as computer recordable storage media.

Alternatively, program code 116 may be transferred to data processingsystem 100 from computer readable media 118 through a communicationslink to communications unit 110 and/or through a connection toinput/output unit 112. The communications link and/or the connection maybe physical or wireless in the illustrative examples. The computerreadable media also may take the form of non-tangible media, such ascommunications links or wireless transmissions containing the programcode.

The different components illustrated for data processing system 100 arenot meant to provide architectural limitations to the manner in whichdifferent embodiments may be implemented. The different illustrativeembodiments may be implemented in a data processing system includingcomponents in addition to or in place of those illustrated for dataprocessing system 100. Other components shown in FIG. 1 can be variedfrom the illustrative examples shown.

For example, a bus system may be used to implement communications fabric102 and may comprise one or more buses, such as a system bus or aninput/output bus. Of course, the bus system may be implemented using anysuitable type of architecture that provides for a transfer of databetween different components or devices attached to the bus system.Additionally, a communications unit may include one or more devices usedto transmit and receive data, such as a modem or a network adapter.Further, a memory may be, for example, memory 106 or a cache such asfound in an interface and memory controller hub that may be present incommunications fabric 102.

The illustrative embodiments described herein provide for a method,computer usable program code and data processing system for performing acalculation using a coordinate rotation digital computer (CORDIC)algorithm. Execution of the CORDIC algorithm is begun. An errorintroduced by a truncated vector as a result of executing the CORDICalgorithm is pre-computed. The error is incorporated into a subsequentiteration of the CORDIC algorithm. Execution of the CORDIC algorithm iscompleted. The result of the CORDIC algorithm is stored.

FIG. 2 is a flowchart of a prior art CORDIC algorithm in which aspectsof the present invention may be implemented. The process shown FIG. 2can be implemented in a data processing system, such as data processingsystem 100 shown in FIG. 1. Specifically, the process shown in FIG. 2can be implemented using a processor.

The standard CORDIC algorithm shown in FIG. 2 uses a set of threeequations to iteratively rotate, from i=1 . . . N, a vector by a givenangle. The form of these equations are given below, where X₀, Y₀ is theoriginal vector and Z₀ is the initial angle.X _(i+1) =X _(i)−/+σ_(j) Y _(i)*2^(−i)  (1)Y _(i+1) =Y _(i)+/−σ_(j) X _(i)*^(2−i)  (2)Z _(i+1) =Z _(i)−σ_(j) tan⁻¹(2⁻¹)  (3)Note that the terms “−/+” and “+/−” are used deliberately in order toindicate that the value of the sign depends on the sign of Z_(i+1).

The only arithmetic operations required to calculate the new vectorcoordinates, X_(i) and Y_(i), are shifts and adds of the previous vectorcoordinates. The new values of the angle Z_(i) are calculated by summingstored values of tan⁻¹(2^(−i)). The value of σ_(i) depends on theresidual angle. If Z_(j)μ0, then σ_(j)=1; if Z_(j)<0, then σ_(j)=−1.

In the process shown in FIG. 2, the subsequent vector {X_(i),Y_(i)} iscalculated by adding the current value of the vector to the shiftedvalue of the opposite coordinate value. For each subsequent iteration,the {X_(i),Y_(i)} values are shifted by I bit positions. In a finiteprecision processor with N fractional bits, there are N−2 additions thatcan potentially introduce truncated error.

The process begins as the processor sets X₀=1, Y₀=0, Z₀=the initialangle, and Phi as having an initial sign of the angle (either −1 or +1)(step 200). The processor then determines if Phi is positive (step 202).If Phi is positive, then the processor calculates a current iteration ofequations (1), (2), and (3) in the form of:X _(i+1) =X _(i) −Y _(i)*2^(−i)  (4)Y _(i+1) =Y _(i) +X _(i)*^(2−i)  (5)Z _(i+1) =Z _(i)+{acute over (α)}  (6)wherein {acute over (α)}=−σ_(j) tan⁻¹(2⁻¹) (step 204). However, if Phiis negative in step 202 (a “no” result), then the processor calculates acurrent iteration of equations (1), (2), and (3) in the form of:X _(i+1) =X _(i) +Y _(i)*2^(−i)  (7)Y _(i+1) =Y _(i) −X _(i)*^(2−i)  (8)Z _(i+1) =Z _(i)+{acute over (α)}  (9)wherein {acute over (α)}=−σ_(j) tan⁻¹(2⁻¹) (step 206).

From either step 204 or 206, the processor determines whether the valueof “i” has reached the value of “N,” which is a number representing thefinal iteration (step 208). If the final iteration has not been reached(a “no” result to step 208), then the processor increments the value of“i” and sets Phi=sign(Z_(i)) (step 210). The process then returns tostep 202 and repeats. However, if the final iteration has been reached(a “yes” result to step 208), then the process terminates.

FIG. 3 is a flowchart of an improved CORDIC algorithm, in accordancewith an illustrative embodiment. The process shown in FIG. 3 can beimplemented in a data processing system, such as data processing system100 shown in FIG. 1. Specifically, the process shown in FIG. 3 can beimplemented using a processor.

As often occurs during execution of the CORDIC algorithm, the truncatedvalues would have generated a carry into the Unit of Least Position ofthe intermediate stage; however, this accuracy is lost with thetraditional CORDIC algorithm shown in FIG. 3. The illustrativeembodiments modify the CORDIC equations (1), (2), and (3) to take intoconsideration these error conditions without adding delay to the overallcalculation.

The methods of the illustrative embodiments increase the effective widthof an N-bit finite precision CORDIC sequence by pre-computing the errorintroduced by the truncated vector, which is denoted as the error bias(Eb). In the modified equations that follow, the “&” symbol is used toindicate the logical AND operation:X _(i+1) =X _(i) −/+Y _(i)*2^(i)+(Y _(i) [i−1]&Y _(i+1) [i−2])  (10)Y _(i+1) =Y _(i) !X _(i)*2^(i)+(X _(i) [i−1]&X _(i+1) [i−2])  (11)The symbols “−/+” and “+/−” are used deliberately to indicate that X_(i)is opposite of Y_(i).

With this modification to the CORDIC equations, the effective width ofthe fixed-width vector is increased to N+1. This illustrative examplecan also be extended past one bit to multiple bit positions forincreased accuracy. The illustrative methods have been verified for N+1and N+2.

An advantage of the illustrative examples is that the error bias can bepre-computed during the computation of the Y_(i) and X_(i) terms so thatno additional delay is added to the carry propagation path that is thecritical path of the CORDIC iterative equations. Another advantage ofthe illustrative examples is that the inherent error in the CORDICalgorithm can be reduced, possibly by 50% or more. In other words, notonly does the CORDIC algorithm of the illustrative embodiments increasethe effective width of an adder, which thereby improves the accuracy ofthe CORDIC calculation, the illustrative embodiments do so at no delaypenalty.

Avoidance of delay penalties is accomplished by pre-computation of anE_(b) term, as defined in equations (12) and (13). The X_(eb) and Y_(eb)equations are as follows:X _(eb) =X _(i)[0]&Y _(i) [i−1]  (12)Y _(eb) =Y _(i)[0]&X _(i) [i−1]  (13)

In every instance, except for the last iteration, the Y_(i)[i−1] andX_(i)[i−1] terms are calculated before the carry propagation completes.This calculation is shown in the flowchart of FIG. 3.

As stated above, for the case where the iteration is on the last loop,the X_(i)[i+1] and the Y_(i)[i+1] terms will arrive shortly after thecomplete carry propagation. In this case, the final error bias is notneeded because no further iterations will occur. Thus, the critical pathis always the carry propagation through the sum of the traditionalCORDIC equations.

In the process shown in FIG. 3, the error bias is computed and added ateach iteration of CORDIC. However, error bias is not computed and addedat the last iteration because the computation and addition are no longeruseful.

In any case, the process begins as the processor sets X₀=1, Y₀=0, Z₀=theinitial angle, X_(eb)=0 and Y_(eb)=0. and Phi as having an initial signof the angle (either −1 or +1) (step 300). The processor then determinesif Phi is positive (step 302).

If Phi is positive (a “yes” result to step 302), then the processorcalculates a current iteration of equations (1), (2), and (3) in theform of:X _(i+1) =X _(i) −Y _(i)*2^(−i) +Y _(ebi)  (14)Y _(i+1) =Y _(i) +X _(i)*^(2−i) +X _(ebi)  (15)Z _(i+1) =Z _(i)+{acute over (α)}  (16)wherein {acute over (α)}=−σ_(j) tan⁻¹(2⁻¹) (step 304).

The processor then determines whether the value of “i” has reached thevalue of “N,” which is the number of the last iteration (step 306). If“yes,” then the process terminates. If “no,” then the processorcalculates equations (17) and (18) below (step 308).X _(eb+1) =X _(i+1) [i−1]&X _(i) [i−2]  (17)Y _(eb+1) =Y _(i+1) [i−1]&Y _(i) [i−2]  (18)

In an illustrative example, the following iteration takes place:

-   I=0: X[0]=1, Y[0]=0-   I=1: X[1]=X[0]−(Y[0]>>i), Y[1]=Y[0]+(X[0]>>i)-   I=2: X[2]=X[1]−(Y[1]>>i), Y[2]=Y[1]+(X[1]>>i)    Note that the plus or minus sign depends on the sign of the Z value,    and is assumed positive for simplification of the example.

Thus, in each iteration, starting at I=1, there is a term that isshifted to the right by one bit. After a number of shifts occur, therecould potentially be a carry generated into the least significant bit.In this illustrative example, the Xeb and Yeb terms are a representationof that carry value. Hence, continuing the example:

-   X[1]=00111 and Y[1]=00101-   X[2]=00111| and Y[2]=00101|−00010|_(—)1_(—)+00011|“1”-   x[3]=00101 and 01000 −00010|_(—)00_ and +00001|“01”

Thus, in this case, EbX(i=3) is the sum (retaining bit positions) of thenumbers separated by underscore symbols (_); hence, 10+00=10. Similarly,EbY(i=3) is the sum of the numbers separated by quotation marks (“ ”);hence, 10+01=11.

As can be seen, there will be cases where there is a carry generatedinto the least significant bit of the X or Y term (or the Unit of LeastPosition of the X/Y terms)

Thus, the Eb term can be represented in multiple forms. One form is forthe current Eb value which would be the Unit of Least Position of theY[i−1] and the Unit of Least Position of the Y[i−2] term; OR the currentEb value can be represented as the error bias from the last X term“anded” with the error bias from the X two iterations in the past.

The processor then increments the value of “i” and changes the sign ofPhi=sign(Z_(i)) (step 310). The process then returns to step 302 andrepeats.

Returning to step 302, if Phi is negative (a “no” result to step 302),then the processor calculates a current iteration of equations (1), (2),and (3) in the form of:X _(i+1) =X _(i) +Y _(i)*2^(−i) +Y _(ebi)  (19)Y _(i+1) =Y _(i) −X _(i)*^(2−i) +X _(ebi)  (20)Z _(i+1) =Z _(i)+{acute over (α)}  (21)wherein {acute over (α)}=−σ_(j) tan⁻¹(2⁻¹) (step 312). The processorthen determines whether the value of “i” has reached the value of “N,”which is the number of the last iteration (step 314). If “no,” then theprocessor calculates equations (22) and (23) below (step 316).X _(eb+1) =X _(i+1) [i−1]&X _(i) [i−2]  (22)Y _(eb+1) =Y _(i+1) [i−1]&Y _(i) [i−2]  (23)

The processor then returns to step 310 and repeats. However, if thevalue of “i” has reached the value of “N” at step 314, then the processterminates.

FIG. 4 is a table comparing errors generated using a prior art CORDICalgorithm versus using the improved CORDIC algorithm shown in FIG. 3, inaccordance with an illustrative embodiment. The table shown in FIG. 4can be generated by a data processing system and stored in acomputer-usable medium.

The illustrative embodiments have been implemented using the C++programming language for its ease of portability and programmingflexibility. Functions were defined for shifting, adding, andsubtracting x and y values for both 32 and 23 bit numbers. Once theactual CORDIC algorithm results were calculated, the x and y values weretruncated by masking off the unwanted bits, which models the finiteprecision hardware. After the truncated results were calculated, asecond mask value was used and modified iteratively to mask off the bitsonce again, but this time retaining an extra bit of precision for eachrun. Comparisons were then done to calculate error in terms of the Unitof Least Position (ULP) difference compared to the untruncated case, anddata was collected as shown in FIG. 4.

Test cases were run on randomly chosen angles between the ranges of −90to +90. The data used for analysis are the full precision CORDICalgorithm X and Y values compared against the 23 and 32 bit truncatedversions as well as the modified CORDIC algorithm results. Error biaswas calculated for 1ULP and 2ULP modified CORDIC algorithm, which isdefined as digits to the right of the Least Significant Bit (LSB) thatare lost in the traditional CORDIC algorithm by the shifting process. Inthese illustrative examples, the Least Significant Bit is the lastnumber in a string of numbers. The modified results significantlyreduced the error as compared to the traditional results for both theaverage and worse case error.

In particular, table 400 of FIG. 4 shows the results of the simulationfor the pre-normalization case of the X_(i) term over two million randomrotation angle calculations. The program was run for 32 and 23 bitcycles in the following set: {29, 30, 31, 32, 33} and {20, 21, 22, 23,24} respectively. Column three shows the results of the truncatedmethod. Column four shows the results for the 1-ULP modified case andcolumn five shows the results for the 2-ULP modified case. As can beseen, the method of the illustrative embodiments offers over 50%improvement in the worst-case error for both the X_(i) and Y_(i) terms.

FIG. 5 is a table comparing errors generated using a prior art CORDICalgorithm versus using the improved CORDIC algorithm shown in FIG. 4, inaccordance with an illustrative embodiment. The table shown in FIG. 5can be generated by a data processing system and stored in acomputer-usable medium.

Table 500 of FIG. 5 is similar to table 400 shown in FIG. 4. Table 500shows the results for the Y_(i) term in the pre-normalization case. Aswith the X_(i) term, the methods of the illustrative embodimentssignificantly improve the error for the Y_(i) term.

In summary, the modified CORDIC algorithm of the illustrativeembodiments reduces the error exhibited by the traditional CORDICalgorithm by adding in pre-calculated error bias into the intermediatesteps of CORDIC algorithm. This illustrative embodiment can also beapplied to most modern CORDIC algorithm implementations such as in highthroughput applications. As a result, the illustrative embodimentsdescribed herein increase both speed and accuracy of CORDIC algorithms.

The invention can take the form of an entirely hardware embodiment, anentirely software embodiment or an embodiment containing both hardwareand software elements. In a preferred embodiment, the invention isimplemented in software, which includes but is not limited to firmware,resident software, microcode, etc. In a preferred embodiment, theinvention is implemented in hardware. In a specific illustrativeembodiment the invention is implemented in a processor or as part of aprocessor. In another specific illustrative embodiment, the invention isimplemented as a separate circuit or chip connected to a bus of a dataprocessing system.

Furthermore, the invention can take the form of a computer programproduct accessible from a computer-usable or computer-readable mediumproviding program code for use by or in connection with a computer orany instruction execution system. For the purposes of this description,a computer-usable or computer readable medium can be any tangibleapparatus that can contain, store, communicate, propagate, or transportthe program for use by or in connection with the instruction executionsystem, apparatus, or device.

The medium can be an electronic, magnetic, optical, electromagnetic,infrared, or semiconductor system (or apparatus or device) or apropagation medium. Examples of a computer-readable medium include asemiconductor or solid state memory, magnetic tape, a removable computerdiskette, a random access memory (RAM), a read-only memory (ROM), arigid magnetic disk and an optical disk. Current examples of opticaldisks include compact disk-read only memory (CD-ROM), compactdisk-read/write (CD-R/W) and DVD.

A data processing system suitable for storing and/or executing programcode will include at least one processor coupled directly or indirectlyto memory elements through a system bus. The memory elements can includelocal memory employed during actual execution of the program code, bulkstorage, and cache memories which provide temporary storage of at leastsome program code in order to reduce the number of times code must beretrieved from bulk storage during execution.

Input/output or I/O devices (including but not limited to keyboards,displays, pointing devices, etc.) can be coupled to the system eitherdirectly or through intervening I/O controllers.

Network adapters may also be coupled to the system to enable the dataprocessing system to become coupled to other data processing systems orremote printers or storage devices through intervening private or publicnetworks. Modems, cable modem and Ethernet cards are just a few of thecurrently available types of network adapters.

The description of the present invention has been presented for purposesof illustration and description, and is not intended to be exhaustive orlimited to the invention in the form disclosed. Many modifications andvariations will be apparent to those of ordinary skill in the art. Theembodiment was chosen and described in order to best explain theprinciples of the invention, the practical application, and to enableothers of ordinary skill in the art to understand the invention forvarious embodiments with various modifications as are suited to theparticular use contemplated.

1. A computer-implemented method for using a result of a calculationthat was performed using a coordinate rotation digital computer (CORDIC)algorithm in digital signal processing, the method comprising the stepsof: beginning execution, by using a processor, of the CORDIC algorithm;pre-computing, by using the processor, an error introduced by atruncated vector as a result of executing the CORDIC algorithm, whereinthe error is pre-computed during the computation of output terms of theCORDIC algorithm, and wherein no delay is added to the carry propagationpath; incorporating, by using the processor, the error into a subsequentiteration of the CORDIC algorithm, wherein the error is incorporatedinto each iteration of the CORDIC algorithm except a first iteration ofthe CORDIC algorithm and a last iteration of the CORDIC algorithm;completing execution, by using the processor, of the CORDIC algorithm;storing, by using the processor, the result of the CORDIC algorithm in amemory; using the result of the CORDIC algorithm in digital signalprocessing; wherein the mathematical equations representing the CORDICalgorithm are modified to have the form:X _(i)+1=X _(i) −/+Y _(i)*2^(i)+(Y _(i) [i−1]&Y _(i+1) [i−2])Y _(i+1) =Y _(i) −/+X _(i)*2^(i)+(X _(i) [i−1]&X _(i+1) [i−2]) wherein“X” and “Y” represent positions in Cartesian coordinates, where “i”represents an iteration number, and the symbol “&” represents a logicalAND operation, wherein X_(i+1) and Y_(i+1) are the output terms of theCORDIC algorithm, wherein “+/−” is an addition or subtraction operation,and wherein “−/+” is a subtraction or addition operation; and whereinthe error is calculated using (Y_(i)[i−1] & Y_(i+1)[i−2]) and(X_(i)[i−1] & X_(i+1)[i−2]).
 2. The computer-implemented method of claim1 wherein a word length of a number is represented by the symbol “N”,and wherein an effective width of a fixed-width vector is increased toN+1.
 3. The computer-implemented method of claim 2 wherein an effectivewidth of a fixed-width vector is increased to N+M, wherein “M” is anumber greater than one.
 4. A computer program product comprising: anon-transitory computer usable medium having computer usable programcode for using a result of a calculation that was performed using acoordinate rotation digital computer (CORDIC) algorithm in digitalsignal processing, the computer program product including: computerusable program code for beginning execution, by using a processor, ofthe CORDIC algorithm, wherein the CORDIC algorithm is used for digitalsignal processing; computer usable program code for pre-computing, byusing the processor, an error introduced by a truncated vector as aresult of executing the CORDIC algorithm, wherein the error ispre-computed during the computation of output terms of the CORDICalgorithm, and wherein no delay is added to the carry propagation path;computer usable program code for incorporating, by using the processor,the error into a subsequent iteration of the CORDIC algorithm, whereinthe error is incorporated into each iteration of the CORDIC algorithmexcept a first iteration of the CORDIC algorithm and a last iteration ofthe CORDIC algorithm; computer usable program code for completingexecution, by using the processor, of the CORDIC algorithm; computerusable program code for storing, by using the processor, the result ofthe CORDIC algorithm in a memory; computer usable program code for usingthe result of the CORDIC algorithm in digital signal processing; whereinthe mathematical equations representing the CORDIC algorithm aremodified to have the form:X _(i)+1=X _(i) −/+Y _(i)*2^(i)+(Y _(i) [i−1]&Y _(i+1) [i−2])Y _(i+1) =Y _(i) −/+X _(i)*2^(i)+(X _(i) [i−1]&X _(i+1) [i−2]) wherein“X” and “Y” represent positions in Cartesian coordinates, where “i”represents an iteration number, and the symbol “&” represents a logicalAND operation, wherein X_(i+1) and Y_(i+1) are the output terms of theCORDIC algorithm, wherein “+/−” is an addition or subtraction operation,and wherein “−/+” is a subtraction or addition operation; and whereinthe error is calculated using (Y_(i)[i−1] & Y_(i+1)[i−2]) and(X_(i)[i−1] & X_(i+1)[i−2]).
 5. The computer program product of claim 4wherein a word length of a number is represented by the symbol “N”, andwherein an effective width of a fixed-width vector is increased to N+1.6. The computer program product of claim 5 wherein an effective width ofa fixed-width vector is increased to N+M, wherein “M” is a numbergreater than one.
 7. A data processing system for using a result of acalculation that was performed using a coordinate rotation digitalcomputer (CORDIC) algorithm in digital signal processing, the dataprocessing system comprising: a bus; at least one processor coupled tothe bus; a computer usable medium coupled to the bus, wherein thecomputer usable medium contains a set of instructions for performing acalculation using a coordinate rotation digital computer (CORDIC)algorithm, wherein the CORDIC algorithm is used for digital signalprocessing, wherein the at least one processor is adapted to carry outthe set of instructions to: begin execution, by using the at least oneprocessor, of the CORDIC algorithm; pre-compute, by using the at leastone processor, an error introduced by a truncated vector as a result ofexecuting the CORDIC algorithm, wherein the error is pre-computed duringthe computation of output terms of the CORDIC algorithm, and wherein nodelay is added to the carry propagation path; incorporate, by using theat least one processor, the error into a subsequent iteration of theCORDIC algorithm, wherein the error is incorporated into each iterationof the CORDIC algorithm except a first iteration of the CORDIC algorithmand a last iteration of the CORDIC algorithm; complete execution, byusing the at least one processor, of the CORDIC algorithm; store, byusing the at least one processor, the result of the CORDIC algorithm ina memory; use the result of the CORDIC algorithm in digital signalprocessing; wherein the mathematical equations representing the CORDICalgorithm are modified to have the form:X _(i)+1=X _(i) −/+Y _(i)*2^(i)+(Y _(i) [i−1]&Y _(i+1) [i−2])Y _(i+1) =Y _(i) −/+X _(i)*2^(i)+(X _(i) [i−1]&X _(i+1) [i−2]) wherein“X” and “Y” represent positions in Cartesian coordinates, where “i”represents an iteration number, and the symbol “&” represents a logicalAND operation, wherein X_(i+1) and Y_(i+1) are the output terms of theCORDIC algorithm, wherein “+/−” is an addition or subtraction operation,and wherein “−/+” is a subtraction or addition operation; and whereinthe error is calculated using (Y_(i)[i−1] & Y_(i+1)[i−2]) and(X_(i)[i−1] & X_(i+1)[i−2]).
 8. The data processing system of claim 7wherein a word length of a number is represented by the symbol “N”, andwherein an effective width of a fixed-width vector is increased to N+1.9. The data processing system of claim 8 wherein an effective width of afixed-width vector is increased to N+M, wherein “M” is a number greaterthan one.